The g-Drazin inverse involving power commutativity

Abstract

Let $\mathcal{A}$ be a complex Banach algebra. An element $a\in \mathcal{A}$ has g-Drazin inverse if there exists $b\in \mathcal{A}$ such that $$b=bab, ab=ba, a-a^2b\in A^{qnil}.$$ Let $a,b\in \mathcal{A}^{d}$. If $a^3b=ba, b^3a=ab,~\mbox{and}~a^2a^db=aa^dba$, we prove that $a+b\in \mathcal{A}^{d}$ if and only if
$1+a^{d}b\in \mathcal{A}^{d}.$ We present explicit formula for $(a+b)^d$ under certain perturbations. These extend the main results of Wang, Zhou and Chen (Filomat, {\bf 30}(2016), 1185--1193) and Liu, Xu and Yu (Applied Math. Comput., {\bf 216}(2010), 3652--3661).